Latin and Greek are both languages of the inflectional type, that is, they use a lot of bound morphology to indicate much of the grammatical information in the language. (In contrast, English and many other languages of the world primarily use syntactic constructions, i.e. combinations of words; the role of bound morphology for grammar is then proportionally smaller.)
Latin had grammatical systems in which both the nouns and the verbs (and to a certain extent the adjectives) fell into classes. The class a word belonged to determined the particular inflectional endings it occurred with
Latin had two simultaneously operative noun class systems: gender (masculine, feminine and neuter) and also what are called declensions. There were five different noun declensions, and the declensions were more important for determining the endings on nouns than the gender. (Declensions gradually became less important and finally essentially disappeared, leaving only the two-way gender classifications in the modern Romance languages). Membership in a given declension was arbitrary, or rather only understandable historically.
The form of a noun that was most diagnostic of which declension it belonged to was the genitive (possessive) case. In other words, you could tell what class the word was in by looking at the genitive form. The declensions were numbered arbitrarily (first through fifth) by the Roman grammarians.
For example, the first declension is identifiable by the ending -ae in the genitive:
The second declension has -i in the genitive:
The third declension has -is in the genitive:
The fourth declension has -us (having a long form of the vowel in the ending) in the genitive:
The fifth declension has -ei in the genitive:
Most nouns fell into one of the first three declensions; the fourth and fifth were rarer.
Each of the declensions had its own set of case and number endings (although there was some degree of overlap). The genitive endings are most important, not only from the Latin point of view (because they indicate which declension a noun belongs to), but from the ENGLISH point of view. The genitive shows the stem of the noun, that is, is the fullest form to which case/number endings were added; and the stem is the form that occurs in almost all English borrowings from Latin.
For example, the word for 'king' is in the nominative case form rex, but its genitive form is regis. Take off the genitive ending -is and we have the stem, reg- , which occurs in Latin borrowings like regicide and regency.
The verbs of Latin fell into four classes, called conjugations. The endings for person and number were slightly different depending on the conjugation; the tense endings show a little more distinction between the classes. The most obvious way the conjugations differed was in the vowel in the second syllable of the root (or first, if only one syllable). That vowel recurred in different forms of the verb, but it is easiest to notice in the infinitive form. (The infinitive is the form that can stand alone, but is not inflected with any person and number endings. (We parse the word in-finit-ive 'without endings'.) The infinitive most closely corresponds translationally to the English to form of a verb as in to go . Notice:
The first conjugation has ain the infinitive:
The second has a long ein the infinitive:
The third has a short e:
And the fourth has an i in the infinitive:
The modern languages descended from Latin have for the most part reduced this to a three-way verb conjugation system, having lost the vowel length distinctions that Latin had. Again, membership in the conjugation classes was essentially arbitrary. It resulted from historical origin and sound change factors, but had nothing to do with the meanings of the verbs.
Knowing just four forms of a verb, one could derive all of the dozens of person/number/tense/voice/mood forms for that verb. These four forms are called the principal parts:
For the last one, you can call it the past participle like its equivalent in English. The "passive" is in there because the meaning of this form is a final state resulting from somebody else's action. For example duct means 'in a state of having been led (by somebody)', just like English opened in It was opened (by someone).
The four principal parts above represent the four basic stems of a Latin verb. Think of Latin word building as follows: A word consists of a ROOT, plus, perhaps, some additional material from a declensional class or conjugation class; together these form a STEM. Then, since Latin is an inflectional language, the word still needs to be filled out with specific inflectional endings. For verbs, these include person, number, tense, voice, and mood, and gender if it is a participle. For nouns, inflections include number, gender, and case. Once the inflections are added we have a full, derived WORD.
The most important forms from the point of view of English borrowings are the present infinitive stem and the past participle stem. They are the forms which almost always appear in English borrowings from Latin. Examples:
Pres. infinitive stem:
Past participle stem: